An 8 by 8 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 12 black squares, can be drawn on the checkerboard?
+506 The smallest possible square that contains at least 12 black squares are the 5 by 5 squares; they either contain 12 or 13 squares. Therefore, we just need to find the number of 5 by 5, 6 by 6, 7 by 7, and 8 by 8 squares, since they all contain at least 12 squares.
Number of 8 by 8 squares = 1^2 = 1
Number of 7 by 7 squares = 2^2 = 4
Number of 6 by 6 squares = 3^2 = 9
Number of 5 by 5 squares = 4^2 = 16
Therefore, the total number of squares with at least 12 squares is 1+4+9+16 = 30
